3 Answers
3

I think this is a matter of style. There's no reason internal references to theorems shouldn't be treated the same way as chapters. The grammatical reason for capitalizing them would be that they are specific designators, but it's somewhat of a judgment call as to whether they are. Here, Google Ngrams come in very handy. If you look at whether people capitalize Chapter 3, they usually do:

And they capitalize Theorem 3 a slightly higher percentage of the time.

Google Ngrams also shows that very few people would capitalize page 3.

@Martin: I expect the use of TeX caught on gradually, between 1980 and 1990. There's a fairly steep upward slope in the use of the capitalized Theorem during that period, which may correspond to this.
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Peter Shor Jun 3 '11 at 18:24

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@Peter: Okay, maybe I don't know how the rules in English are made but to me saying that it is correct because more people use it is an ad populum fallacy.
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Jonas TeuwenJun 4 '11 at 22:38

Theorem 4.5 is the title of the item in your book. You would refer to it in the same way that you would refer to a chapter or section of the book, or the name of another book, by reproducing the same capitalization as the title's original appearance.

It looks unusual because we are used to making other references within the same book by chapter number (or page number, or whatever), rather than by title. This is purely a matter of convenience; you have a general idea of where "chapter two" is where you'd probably have to look up "Elliptical Integrals". It just happens that in this case the title contains a number.

Very nice answer. Indeed, we capitalize Table 4.3 because it's a title. You can see this with Google Ngrams, by checking that 'See Table' is much more common than 'See table' (usually part of a construction like See Table 4.3); while 'See Table on' is much less common than 'See table on' (usually part of a construction like See table on page 77).
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Peter Shor Jun 3 '11 at 15:38

I don't really understand this answer. If I write "Theorem 4.5." I would capitalize it because it is the first word in "sentence". This is the original appearance of the title, what am I missing?
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Jonas TeuwenJun 3 '11 at 15:54

@Peter: thank you very much for the links; I'm afraid I've never learned how to work Google Ngrams, but it is a good tool for showing this particular distinction!
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user1579Jun 3 '11 at 15:54

@Jonas: sorry, I don't think I was very clear in my haste. By "original appearance" I meant where it occurs, not what it looks like. Think of it as the title of the sub(sub(sub...))section in which the theorem itself appears. Unfortunately I don't have time to rewrite the answer properly at the moment, since my downloads are all finished and I need to get on with some work! I'll see if I can put something better together this evening.
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user1579Jun 3 '11 at 16:00

Why are things like "Theorem 4.5" and so on proper nouns? They are not the same throughout all texts. It is not something like "Pythagoras' Theorem" which is the same througout all texts.
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Jonas TeuwenJun 3 '11 at 12:45

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I must disagree with this answer. One does not say "Go to Page 5." Since it's a simple enumeration, "theorem 1" is really an abbreviation for "the first theorem" or "the theorem [which is] numbered 1."
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QuadrescenceJun 3 '11 at 12:51

@Jonas: "London" (or "George") is not less of a proper noun simply because there are multiple possible referents.
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Ben VoigtJun 5 '11 at 2:34